which are approximations to \(J(u)\)  and \(\nabla J(u)\)  respectively.

Based on the well known Armijo-Wolfe conditions from continuous optimization [1]}, [2]}, [3]}, we introduce the following step size conditions:
 For \(0<c_1<c_2<1\) , we call \(s_n(\tau _n)\)  an Armijo step, if
\(&\tilde{J}_n(s_n(\tau _n)) \le \hat{J}_n - c_1\hat{G}_n^\top \left(u_n - s_n(\tau _n)\right).   \\\multicolumn{2}{l}{\text{Additionally, we define the following Wolfe-type condition:}}\\&\tilde{G}_n(s_n(\tau _n))^\top \left(s_n(\tau _n)-u_n \right) \ge c_2\hat{G}_n^\top \left(s_n(\tau _n)-u_n \right) .  \) 
